To determine which number is a solution of the inequality \( x(7 - x) > 8 \), we will test each of the provided numbers: -1, 2, 8, and 0.
First, we can rewrite the inequality:
\[ x(7 - x) - 8 > 0 \]
This simplifies to:
\[ 7x - x^2 - 8 > 0 \]
or
\[ -x^2 + 7x - 8 > 0 \]
Next, let's find the roots of the quadratic equation \( -x^2 + 7x - 8 = 0 \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = -1 \), \( b = 7 \), and \( c = -8 \):
\[ x = \frac{-7 \pm \sqrt{7^2 - 4(-1)(-8)}}{2(-1)} \]
Calculating the discriminant:
\[ \sqrt{49 - 32} = \sqrt{17} \]
So, the roots are:
\[ x = \frac{7 \pm \sqrt{17}}{2} \]
Now let's test the provided values in the original inequality one by one:
-
For \( x = -1 \): \[ -1(7 - (-1)) = -1(7 + 1) = -1(8) = -8 \] \(-8 > 8\) is false.
-
For \( x = 2 \): \[ 2(7 - 2) = 2(5) = 10 \] \(10 > 8\) is true.
-
For \( x = 8 \): \[ 8(7 - 8) = 8(-1) = -8 \] \(-8 > 8\) is false.
-
For \( x = 0 \): \[ 0(7 - 0) = 0(7) = 0 \] \(0 > 8\) is false.
The only value that satisfies the inequality \( x(7 - x) > 8 \) is:
\[ \boxed{2} \]